- The paper introduces a non-asymptotic framework for identifying LTI system Markov parameters using finite input/output data.
- It analytically derives finite sample complexity bounds that quantify the data needed for achieving desired estimation accuracy.
- The study demonstrates the stability of the Ho-Kalman algorithm and its robustness to noise in practical system identification scenarios.
Non-asymptotic Identification of LTI Systems from a Single Trajectory
The paper "Non-asymptotic Identification of LTI Systems from a Single Trajectory" by Samet Oymak and Necmiye Ozay addresses the problem of estimating a realization of linear time-invariant (LTI) systems using finite data. In particular, the authors focus on determining the sample complexity for identifying system parameters from a single input/output trajectory. This paper is pertinent due to the increasing importance of data-driven methods in control system design where state-space models are impractical to derive from first principles.
Summary and Main Contributions
The authors improve upon traditional asymptotic methods by focusing on the non-asymptotic, finite-time analysis of system identification—a necessity when dealing with real-world data limitations. They present a practical method for estimating the Markov parameters of a system using a single trajectory. The paper provides finite sample complexity results to obtain a balanced realization of the system with desired accuracy using the Ho-Kalman algorithm.
The paper’s contributions include:
- Finite Sample Complexity for Markov Parameters: The authors analytically derive sample complexity bounds for learning the Markov parameters directly from least-squares estimates of input/output trajectories. They show how the amount of data impacts the estimation accuracy, effectively addressing scenarios where data is limited.
- Stability of Learning Algorithms: The paper proves that traditional techniques like the Ho-Kalman algorithm are stable in non-asymptotic settings. These algorithms can estimate system parameters efficiently without infinite data.
- Approximation of System’s Hankel Operator: They approximate the Hankel operator using Markov parameter estimates with minimal data, exploiting system stability to derive near-optimal sample sizes.
Technical Approach
The approach involves crafting a least-squares problem from a single trajectory of input/output pairs, used to estimate the Markov parameters of the system. The authors outline how these parameters aid in reconstructing a balanced realization using spectral methods associated with the Ho-Kalman algorithm. A pivotal analytical tool in their methodology is the use of concentration bounds and stability results specific to circulant matrices—a complex linear algebra structure that accommodates the dependencies inherent in system trajectories.
Robustness and Implications
A significant claim is that their method is robust even in the presence of noise or modeling inaccuracies, as opposed to standard asymptotic methods which assume noise-free conditions. This robustness provides practical implications for real-world applications in areas like autonomous driving or industrial automation where data acquisition is costly and limited.
The implications extend beyond theoretical bounds, paving the way for enhanced data-driven control solutions in modern systems. They provide clear quantification and rigorous guarantees of estimator performance—crucial for iterative control tasks in uncertain environments.
Future Directions
The work sets the groundwork for further exploration into optimization-based techniques that could be more effective than traditional realization algorithms like Ho-Kalman. Exploring constraints like subspace conditions on system matrices could yield more tractable models. Additionally, integration with Markov parameter-based control synthesis could improve the efficiency of model predictive control and robust control frameworks.
Ultimately, the findings of this paper serve as a robust starting point for future investigations into non-asymptotic analysis for LTI systems, highlighting both theoretical and practical pathways to enhance system identification in real-world applications.